Triangulation invariance of statistics for topological excitations

Let $p$ be the excitation dimension, let $G$ be a fusion group, and let $M$ be a triangulable topological space. For any triangulation $C$ of $M$, write $T_p(M,G)$ for the statistics of $p$-dimensional topological excitations on $M$ and $T_p(C,G)$ for the corresponding statistics on $C$.

Triangulation-invariance conjecture. There is a good definition of $T_p(M,G)$, and

$$T_p(M,G) \simeq T_p(C,G).$$

This would make the statistics independent of the chosen triangulation and extend the simplicial construction to triangulable topological spaces. The source explicitly leaves both the definition and the claimed invariance conjectural.